Lesson 11: Too Much Precision
Given all the errors in the models, their unrealistic assumptions, and the frankly bizarre ways in which they are used, it is surprising that banks and funds make money at all!
Actually there is a sound reason why they make money and it has nothing to do with clever models. As mentioned above, if you take all equities and derivatives based on those equities, and sum them up across all banks, hedge funds, individuals, etc., then all the complicated derivatives will cancel themselves out since for every long there is a short, leaving just the equities, plain and simple. Now put most of those equities in the hands of banks and funds, rather than in the hands of the man in the street, and you'll see that as long as the stock market is rising those banks and funds will make money, bonuses all around. All derivatives do is to redistribute the profit, some banks/funds win, some lose, but net no impact. This argument only works during growing markets. In bear markets you need to look elsewhere for profit, ways of making money whether stocks are going up, down or sideways, and that means derivatives.
But derivatives come with a lot of risk, some well understood and some not. And as people get more 'sophisticated' they believe that they can increase their exposure to derivatives and so make money regardless of market conditions. Sometimes this is true, and the models seem to work, sometimes it is not and the models fall over. It's because of potential model error that one has to build in a decent margin, based on a decent understanding of possible risks. But once banks start competing between themselves for the same contracts then the margin for error will inevitably succumb to the powerful forces of supply and demand.
One part of the above paragraph glosses over something that is very important in practice. When I say 'Sometimes this is true, and the models seem to work,' you should ask how do we know? The truth of the matter is that we don't. Let's look at the story of a single trade in some exotic. There are several stages.
1. There is demand for some contract, real or perceived
2. The contract must be understood in terms of risk, valuation, potential market, profit, etc.
3. A deal gets done with an inbuilt profit margin.
4. The contract is then thrown into some big pot with lots of other contracts and they are risk managed together.
5. A profit is accrued, perhaps marking to model, or perhaps at expiration.
This is naturally a simplification but I wanted to list this procedure to highlight a disconnection between theory and practice. At Stage 2 here there is a valuation procedure that will probably involve some complicated model, maths, and numerics. The underlying model makes certain assumptions about the financial world and the contract is valued accordingly. But then at Stage 4 you throw this contract into the same pot as other contracts, and you don't actually implement the theory upon which the valuation was based. And finally, you never really know whether that particular contract made money because it gets hedged simultaneously with other contracts. You never follow the valuation, static and dynamic hedging, with all its cash flows and accrued interest, etc., that the theory is using. Instead you look at the pot containing many contracts and say that it has made a profit (or a loss). You have no idea whether each contract has 'washed its own face' as they say.
I have to ask why bother going to all this trouble of valuation and risk management when you lump everything together anyway? I'm not saying that it is wrong to lump everything together, on the contrary, in the above lesson on nonlinearity you can see sound financial reasons why you should do that. I am saying that such effort in valuation hardly seems worth it when ultimately you are probably going to be making money from the Central Limit Theorem! Errors in the models and the implementation of the models are probably large enough that on a contract-by-contract basis you may make or lose money, but so what? As long as on average you make money then you should be happy.
The point of this lesson is to suggest that more effort is spent on the benefits of portfolios than on fiddly niceties of modelling to an obsessive degree of accuracy. Accept right from the start that the modelling is going to be less than perfect. It is not true that one makes money from every instrument because of the accuracy of the model. Rather one makes money on average across all instruments despite the model. These observations suggest to me that less time should be spent on dodgy models, meaninglessly calibrated, but more time on models that are accurate enough and that build in the benefits of portfolios.
While we are on the topic of model accuracy, I just want to make a few comments about models in different markets.
Some models are better than others. Sometimes even working with not-so-good models is not too bad. To a large extent what determines the success of models is the type of market. Let me give some examples.
Equity, FX and commodity markets
Here the models are only so-so. There has been a great deal of research on improving these models, although not necessarily productive work. Combine less-than-brilliant models with potentially very volatile markets and exotic, non-transparent, products and the result can be dangerous. On the positive side as long as you diversify across instruments and don't put all your money into one basket then you should be ok.
These models are pretty dire. So you might expect to lose (or make) lots of money. Well, it's not as simple as that. There are two features of these markets which make the dire modelling less important; these are (a) the underlying rates are not very volatile and (b) there are plenty of highly liquid vanilla instruments with which to try to hedge model risk. (I say 'try to' because most model-risk hedging is really a fudge, inconsistent with the framework in which it is being used.)
Oh, Lord! Instruments whose pricing requires input of correlation (FI excepted, see above) are accidents waiting to happen. The dynamic relationship between just two equities can be beautifully complex, and certainly never to be captured by a single number, correlation. Fortunately these instruments tend not to be bought or sold in non-diversified, bank-destroying quantities. (Except for CDOs, of course!)
Single-name instruments are not too bad as long as the trades are kept small. Except that often the models assume risk neutrality and an ability to hedge that is often not possible in practice. Again problems arise with any instrument that has multiple 'underlyings,' so the credit derivatives based on baskets... you know who you are. But as always, as long as the trades aren't too big then it's not the end of the world.